Question: Process Dynamics Control Law help.

[specwarop]

Homework Statement

A crude oil preheater furnace has a simplified mathematical model for the process obtained by applying a steady-state energy balance that requires that: Heat input through fuel = heat required to heat feed from T(i) to T, ie:

$$\lambda$$_f.Q_f = F.C_p.(T-T_i)

Here, C_p is the specific heat capacity of the crude oil feed.

a) Use this information to derive a control law that, given measurements of the disturbance variable T_i, and assuming that the physical properties of the fuel and the crude oil are available, prescribes how to adjust Q_f so that T attains the desired value T*.
b) Identify the potential problems a controller based on such a control law is likely.

Letters in brackets are actually subscript.

Homework Equations

Refer equation above.

The Attempt at a Solution

I have attempted this question however I honestly have no idea where to go with it.
I have determined which are input and output variables and which are disturbances. But once I work out the y, u, d deviation variables, I get stuck.
In addition, this above question is from the Ogunnaike book on Process Dynamics, which fails to provide answers in the book, or in an external solution manual. So I am therefore unable to check my answers at all.

Any help appreciated.

 

[KataKoniK]

Hello,

Thank you for your post. I understand that you are struggling with deriving a control law for a crude oil preheater furnace based on the given information. Here are some steps that may help you:

1. Identify the input, output, and disturbance variables: The input variable in this case is Qf, the heat input through fuel. The output variable is T, the desired temperature of the crude oil. The disturbance variable is Ti, the initial temperature of the crude oil.

2. Define the deviation variables: Let y = T-T* be the deviation of the output variable from its desired value. Similarly, let u = Qf-Qf* be the deviation of the input variable from its desired value. And let d = Ti-Ti* be the deviation of the disturbance variable from its desired value.

3. Write the steady-state energy balance equation: The given equation can be rewritten as:

Qf = F*Cp*(T-Ti)/λ

4. Substitute the deviation variables into the equation: Using the deviation variables defined in step 2, the equation becomes:

Qf* + u = F*Cp*(T* + y - (Ti* + d))/λ

5. Simplify the equation: Rearrange the equation to solve for u:

u = λ*F*Cp*(T* + y - (Ti* + d)) - Qf*

6. Substitute the values of λ, F, and Cp: These values are given in the problem statement, so you can substitute them into the equation to get the final control law:

u = K*(T* + y - (Ti* + d)) - Qf*

Where K = λ*F*Cp is a constant that depends on the physical properties of the fuel and crude oil.

Now, for part b, the potential problems with using this control law could include:

1. Accuracy of the physical properties: The control law assumes that the physical properties of the fuel and crude oil are known and accurate. Any errors in these values could affect the performance of the control law.

2. Steady-state vs. dynamic behavior: The control law is based on a steady-state energy balance, which may not accurately capture the dynamic behavior of the system. This could lead to instability or poor control performance.

3. Nonlinearities in the system: The control law assumes a simplified linear model, but in reality, the system may have nonlinearities that can affect the control performance.